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Mirrors > Home > MPE Home > Th. List > lgsabs1 | Structured version Visualization version GIF version |
Description: The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.) |
Ref | Expression |
---|---|
lgsabs1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgscl 25889 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | |
2 | 1 | zcnd 12091 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℂ) |
3 | 2 | abscld 14798 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ∈ ℝ) |
4 | 1re 10643 | . . 3 ⊢ 1 ∈ ℝ | |
5 | letri3 10728 | . . 3 ⊢ (((abs‘(𝐴 /L 𝑁)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ ((abs‘(𝐴 /L 𝑁)) ≤ 1 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁))))) | |
6 | 3, 4, 5 | sylancl 588 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ ((abs‘(𝐴 /L 𝑁)) ≤ 1 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁))))) |
7 | lgsle1 25890 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1) | |
8 | 7 | biantrurd 535 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ (abs‘(𝐴 /L 𝑁)) ↔ ((abs‘(𝐴 /L 𝑁)) ≤ 1 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁))))) |
9 | nnne0 11674 | . . . 4 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ → (abs‘(𝐴 /L 𝑁)) ≠ 0) | |
10 | nn0abscl 14674 | . . . . . . . 8 ⊢ ((𝐴 /L 𝑁) ∈ ℤ → (abs‘(𝐴 /L 𝑁)) ∈ ℕ0) | |
11 | 1, 10 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ∈ ℕ0) |
12 | elnn0 11902 | . . . . . . 7 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ0 ↔ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ∨ (abs‘(𝐴 /L 𝑁)) = 0)) | |
13 | 11, 12 | sylib 220 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ∨ (abs‘(𝐴 /L 𝑁)) = 0)) |
14 | 13 | ord 860 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (abs‘(𝐴 /L 𝑁)) ∈ ℕ → (abs‘(𝐴 /L 𝑁)) = 0)) |
15 | 14 | necon1ad 3035 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ≠ 0 → (abs‘(𝐴 /L 𝑁)) ∈ ℕ)) |
16 | 9, 15 | impbid2 228 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ (abs‘(𝐴 /L 𝑁)) ≠ 0)) |
17 | elnnnn0c 11945 | . . . . 5 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ0 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁)))) | |
18 | 17 | baib 538 | . . . 4 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ0 → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ 1 ≤ (abs‘(𝐴 /L 𝑁)))) |
19 | 11, 18 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ 1 ≤ (abs‘(𝐴 /L 𝑁)))) |
20 | abs00 14651 | . . . . . 6 ⊢ ((𝐴 /L 𝑁) ∈ ℂ → ((abs‘(𝐴 /L 𝑁)) = 0 ↔ (𝐴 /L 𝑁) = 0)) | |
21 | 20 | necon3bid 3062 | . . . . 5 ⊢ ((𝐴 /L 𝑁) ∈ ℂ → ((abs‘(𝐴 /L 𝑁)) ≠ 0 ↔ (𝐴 /L 𝑁) ≠ 0)) |
22 | 2, 21 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ≠ 0 ↔ (𝐴 /L 𝑁) ≠ 0)) |
23 | lgsne0 25913 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 /L 𝑁) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1)) | |
24 | 22, 23 | bitrd 281 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1)) |
25 | 16, 19, 24 | 3bitr3d 311 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ (abs‘(𝐴 /L 𝑁)) ↔ (𝐴 gcd 𝑁) = 1)) |
26 | 6, 8, 25 | 3bitr2d 309 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 ≤ cle 10678 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 abscabs 14595 gcd cgcd 15845 /L clgs 25872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-prm 16018 df-phi 16105 df-pc 16176 df-lgs 25873 |
This theorem is referenced by: lgssq 25915 lgssq2 25916 lgsquad3 25965 |
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